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The Probability Density Function (PDF) is a fundamental concept in probability and statistics. It describes how the probability of a continuous random variable is distributed over its possible values. In simpler terms, the PDF tells you the likelihood of a random variable taking on a particular value in a continuous range. For the gamma distribution, the PDF is defined using the shape parameter \(\alpha\) and the scale parameter \(\beta\). The formula looks like this: \[ f_X(x) = \frac{x^{\alpha-1} e^{-x/\beta}}{\beta^\alpha \Gamma(\alpha)} \]where \(\Gamma(\alpha)\) is the gamma function. In the exercise, we transformed a random variable from hours to minutes, leading to the formation of a new PDF, \(f_Y(y)\), by relation: \[ f_Y(y) = f_X\left(\frac{y}{60}\right) \cdot \frac{1}{60} \].Here, we used the chain rule of differentiation to obtain the PDF of \(Y\). This step is pivotal because it transforms the PDF from its original unit to the new unit while preserving the probability structure.

The Cumulative Distribution Function (CDF) represents the probability that a random variable takes on a value less than or equal to a certain number. For a gamma-distributed variable \(X\), this would be expressed as \(F_X(x) = P(X \leq x)\). The CDF is beneficial because it gives a sense of the cumulative probability accumulation up to a point. To find the CDF of the transformed variable \(Y = 60X\), we used the transformation hint: \(Y \leq y\) if and only if \(X \leq y/60\). Therefore, the CDF of \(Y\) is \(F_Y(y) = F_X(y/60)\), allowing us to connect the random variable in hours to minutes. This CDF provides the basis for further finding the PDF through differentiation, linking continuous probability across transformations.

Transforming random variables is an essential operation when dealing with different measurement scales or units. In this exercise, the lifetime of a component measured in hours was transformed to minutes. This is mathematically represented by \(Y = 60X\), where \(X\) represents the original random variable and \(60\) is the transformation constant. Such transformations allow us to express probabilities in the new units of measure while adhering to original statistical properties. When transforming distributions like the gamma distribution, not only do we need to adjust the values, but also consider how kernels such as the PDF and CDF behave. By differentiating the transformed CDF, we elegantly derived the new PDF for \(Y\). This shows us how versatile probability distributions are through proper transformations.

The scale parameter \(\beta\) is a critical element of the gamma distribution, which affects the spread or scaling of the distribution's PDF and CDF. The scale parameter essentially stretches or compresses the distribution along the x-axis. When we transform a random variable like \(X\) with a gamma distribution and change it by a constant \(c\), creating \(Z = cX\), the new gamma distribution of \(Z\) sees its scale parameter adjusted to \(\beta/c\). This means the scale parameter is inversely proportional to the constant \(c\)—dividing \(\beta\) by \(c\) stretches the distribution wider, indicating a longer spread of values. Understanding the scale parameter is crucial for making precise probability assessments in different transformational scenarios, keeping the probabilistic behavior intact although in a different measure.